1. Field of the Invention
This invention pertains generally to probability density estimation, and more particularly to probability density estimation in imagery.
2. Description of Related Art
High resolution and hyperspectral satellite images, city and county boundary maps, census data, and other types of geographical data provide much information about a given region. It is desirable to integrate this knowledge into models defining geographically dependent data.
Given spatial event data, a probability density may be constructed that estimates the probability that an event will occur in a region. Often, it is unreasonable for events to occur in certain regions, and thus it is ideal for the model to reflect this restriction. For example, residential burglaries and other types of crimes are unlikely to occur in oceans, mountains, and other regions. Such areas can be determined using aerial images or other external spatial data, and these improbable locations are generally denoted as an invalid region. Ideally, the support of all density data should therefore be contained in the valid region.
Geographic profiling, a related topic, is a technique used to create a probability density from a set of crimes by a single individual to predict where the individual is likely to live or work. Some law enforcement agencies currently use software that makes predictions in unrealistic geographic locations. Methods that incorporate geographic information have recently been proposed and is an active area of research.
A common method for creating a probability density is to use Kernel Density Estimation, which approximates the true density by a sum of kernel functions. A popular choice for the kernel is the Gaussian distribution which is smooth, spatially-symmetric, and has non-compact support. Other probability density estimation methods include the taut string, logspline, and the Total Variation Maximum Penalized Likelihood Estimation models. However, none of these methods utilize information from external spatial data. Consequently, the density estimate typically has some nonzero probability of events occurring in the invalid region.